Math E-320: Fall 2015
Teaching Math with a Historical Perspective
Mathematics E-320:
Instructor: Oliver Knill
Office: SciCtr 432

Lecture 13: Computing

In this last lecture, we will look at the history of computing. After saying a bit more about the project, we will look at the beginnings of computing (abacus, Antikythera, early computing machines until the modern computers). A NYT article on the Antikythera. We will also take the opportunity to point out changes in how we can do mathematics like for example, how to use computers to experiment or artificial intelligence to prove things. An other topic we will glance over are the limits of computing. There are problems called NP hard problems which appear to be difficult. One of the most famous open problems is to understand this and see whether there are indeed problems which can not be solved in polynomial time. There are other limitations like Turings Halte problem. There are fundamental limitations what a computing device can do. Turing came up with the concept of Turing machines. Its not yet clear whether how much quantum computers will make an impact in the future. While quantum computers can not compute more than a Turing machine, there are indications that quantum computers could speed up computations significantly. So far this has not been realized and there could be fundamental limitations. A recent article on Quantum computing. and here an article in German mentioning the article from September 7 about D-Wave. An mathematical difficulty in computing related to quantum mechanics has appeared here on December 10, 2015.

Lecture 12: Dynamics

What happens if one applies a map again and again? In general, it produces an unpredictable sequence of numbers. One calls this chaos. The theory started to grow at the beginning of the 20'th century and bloomed in the last half. Even so the story is pretty well understood, major problems remain to be solved. In particular there is a huge discrepancy of what one measures and what one can prove, even for very simple systems. We will look at the history of this fascinating topic which ranges from celestial mechanics to keep pushing the same button on a calculator.

Lecture 11: Cryptology

Cryptology the science of breaking codes. The journey starts in 2000 BC and goes up to today where encryption is a hot topic. We will look first at basic substitution cyphers like invented by Caesar, then at more sophisticated versions up to the Enigma, which was used in WW2. Then we switch to public key encryption. It is a bit ambitious but we will see how key exchange and sending encrypted data works using primes. This is RSA and Diffie Helleman. Finally, we will also have a glimpse at quantum cryptology.

Lecture 10: Analysis

Analysis is a huge field: complex analysis, functional analysis, operator theory, inverse problems, harmonic analysis, real analysis, calculus of variations, spectral analysis are all part of it. Since it is so large, it is hopeless to get an overview: we will focus almost entirely on one topic, the geometry of fractals. Our goal is to understand one single formula, the formula for computing dimension. So, there will be lots of pictures and of course movies and many examples.

Lecture 9: Topology

In this lecture we look at topology. We first look at topological equivalence which mathematicians call with a fancy name: homeomorphic. It means for example that an apple is topologically equivalent to a pear or that a doughnut is equivalent to a cup. We will practice this on various objects. Much of the time, we will spend with polyhedra and see them both historically as well as in pop culture or philosophy. We will also give the detailed proof that there are exactly 5 platonic solids which is a theorem of Theaetetus. We also will informally look at platonic solids in higher dimensions which have been explored first by less main stream mathematicians like Ludwig Schlaefli and Alicia Boole Stott. The subject of topology is also linked to graph theory, which started with the Koenigsberg bridge problem. Both for analyzing polyhedra and graphs, there is the important notion of Euler characteristic and Euler's gem.

Lecture 8: Probability theory lecture

[ Update of December 1, 2015: Business insider (German) treats 4 problems (2 of which we covered in the lecture). The first is the Monty Hall problem, the second is the boy-girl problem). There are two other famous problems. You have a leaky sink. Somebody helps you to fix it. What is more probable: the person is a Bookkeeper or a Bookkeeper and Plumber? The four problem is the friendship paradox: It is more probable that one of your friends has more friends than you.

] The fact that probability theory can be a bit of a tougher topic to grasp is illustrated by the fact that it was developed relatively late: Mathematicians started to think about it first in the context of gambling. The clear mathematical foundations were only led about 60 years ago with Kolmogorov.

Lecture 7: Set theory lecture

Lecture 6: Calculus lecture

Calculus in 20 minutes:

Lecture 5: Algebra lecture

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Math E320| Oliver Knill | Fall 2015 | Canvas page | Extension School | Harvard University